Uniform pdf from difference of two stochastic variables?

In summary, the conversation discusses the search for a probability distribution (D) that would satisfy the property of having a uniform distribution for the difference of two independent stochastic variables. One suggestion is to use the characteristic function and take the inverse Fourier transform, but the attempt to do so has not been successful. The conversation ends with a suggestion to try getting the distribution function and a recommendation to seek help in a mathematics forum.
  • #1
bemortu
5
0
Hi,

I'm trying to find a probability distribution (D) with the following property:
Given two independent stochastic variables X1 and X2 from the distribution D, I want the difference Y=X1-X2 to have a uniform distribution (one the interval [0,1], say).

I don't seem to be able to solve it. I'm not even sure that such a distribution exists...

Any ideas?
 
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  • #2
Sugestion (outline). For simplicity I will make it uniform on the interval (-1/2,1/2). The characteristic function is sin(t/2)/(t/2). Take the square root and then the inverse Fourier transform should give you something close to what you want (the sum of two random variables will have a uniform distribution).
 
  • #3
Yes, that's one of the things I already tried. The problem is that I didn't manage to calculate that inverse Fourier transform. I tried it with Mathematica, which could not find an analytical solution. I also tried the numerical inverse Fourier transform in Mathematica but it also failed. Maybe it means that this distribution doesn't exist?
 
  • #4
My guess: there is no density function. You might try getting the distribution function.

F(y) - F(x) = 1/2π ∫{(exp(ity) - exp(itx))φ(t)/(it)}dt
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http://mathforum.org/kb/forumcategory.jspa?categoryID=16

You might try the above forum - it is more mathematical.
 
Last edited:
  • #5


I can provide some insight into this problem. The distribution you are describing is known as the "difference distribution", and it is commonly used in statistics and probability theory. This distribution can be derived by taking the convolution of the two original distributions, X1 and X2.

In general, the convolution of two distributions is not always easy to calculate. However, in your case, since you are looking for a uniform distribution, the convolution can be simplified. The convolution of two uniform distributions is also a uniform distribution, so you can simply take the difference between the two uniform distributions to get the desired result.

In conclusion, the distribution you are looking for does exist and can be derived through the convolution of the two original distributions. I would recommend consulting a textbook or a statistician for more detailed information on how to calculate this distribution for your specific case.
 

1. What is a uniform pdf from difference of two stochastic variables?

A uniform pdf from difference of two stochastic variables is a probability density function that describes the distribution of values obtained by subtracting two random variables from each other. It is a continuous distribution with a constant probability density over a specific range of values.

2. How is a uniform pdf from difference of two stochastic variables different from a regular uniform distribution?

A regular uniform distribution has a constant probability density over a fixed range of values, while a uniform pdf from difference of two stochastic variables has a variable range depending on the values of the two random variables being subtracted. This results in a non-constant probability density function.

3. What is the formula for calculating a uniform pdf from difference of two stochastic variables?

The formula for a uniform pdf from difference of two stochastic variables is f(x) = 1/(b-a), where b and a are the upper and lower limits of the range of values obtained by subtracting the two random variables. This assumes that both variables are independent and have equal probability distributions.

4. What is the significance of a uniform pdf from difference of two stochastic variables in scientific research?

A uniform pdf from difference of two stochastic variables is commonly used in research to model the difference between two random variables, such as in the study of reaction rates or measurement errors. It allows for a more accurate representation of the underlying data and can provide insights into the relationship between the two variables.

5. Can a uniform pdf from difference of two stochastic variables have a negative probability density?

No, a uniform pdf from difference of two stochastic variables can never have a negative probability density. This is because the range of values for the difference of two random variables is always positive, resulting in a constant or zero probability density over that range.

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